Use the results of the specified exercises to determine (a) the domain and (b) the range of each

step 1 Here we have the function, Y is equal to the absolute value of X plus 4, and then minus 3. So th

Use the results of the specified exercises to determine (a)...

Key Concepts

Function Transformations

Function transformations include operations such as translations, reflections, and dilations that alter the graph of the original function. In this context, horizontal shifts (affecting the input variable) and vertical shifts (affecting the output value) are crucial. These transformations change the position of the graph in the coordinate plane and are essential in determining the new domain and range of the function.

Absolute Value Function

The absolute value function returns the non-negative magnitude of a number, regardless of its sign. This characteristic ensures that without any modifications, the outputs are always at least zero. Understanding this property is fundamental when determining the behavior of the function, particularly the range, as it indicates that the basic output values will be shifted by any additional constants.

Range

The range of a function consists of all possible output values the function can attain. Determining the range involves analyzing the behavior of the function, particularly how transformations affect the minimum or maximum values. When a function involves the absolute value, which inherently produces only non-negative outputs, any vertical shifts or reflections can modify the starting point of the outputs.

Domain

The domain of a function represents all possible input values for which the function is defined. When analyzing functions, especially those involving basic operations like absolute values, it is important to verify if any restrictions apply to the variable x. In many cases, as with most absolute value functions, there are no limitations on x, meaning the function is defined for every real number.

Transcript

Please give Ace some feedback